Abstract

We consider higher order Schrödinger type operators with nonnegative potentials. We assume that the
potential belongs to the reverse Hölder class which includes nonnegative polynomials. We establish
estimates of the fundamental solution and show boundedness of some Schrödinger type operators.

1. Introduction

Let be a nonnegative potential and consider the Schrödinger type operators on , where is a positive integer and . When is a nonnegative polynomial, Zhong proved estimates of the fundamental solution for and and showed some estimates for and (see [1]). More precisely, he showed the boundedness of the operators , , and , where . He also proved that the operators and are Calderón-Zygmund operators. Recently, in [2], the authors showed the boundedness of the operators , , and for nonnegative polynomial potential .

For the potential which belongs to the reverse Hölder class, which includes nonnegative polynomials, Shen generalized Zhong's results on (see [3]). Actually, he established estimates of the fundamental solution for and showed the estimates for the operators , , , and so on. For the operator with reverse Hölder class potentials, further results have been investigated by many researchers. See [4–7], for example. For the operator with reverse Hölder class potentials, in [8], the author established estimates of the fundamental solution for and showed the boundedness of the operators , where . In [8], the author also showed that the operator is a Calderón-Zygmund operator. Recently, in [9], the authors showed boundedness of the operator and that the operator is a Calderón-Zygmund operator.

In this paper, we study with reverse Hölder class potential on , where is a nonnegative integer and . We establish estimates of the fundamental solution for and show the boundedness of the operators , where is an integer satisfying .

As mentioned above, in [2], the authors proved some results on , where is a nonnegative polynomial and is an integer, . They proved their results by making use of [2, Lemma 3.3 and Corollary 3.1] which have been proved only for nonnegative polynomial . In this paper, our strategy is different from the one in [2], since the question whether the above two results can be proved for reverse Hölder class potentials is yet to be settled. The purpose of this paper is to show some results on with potential which belongs to the reverse Hölder class, which includes nonnegative polynomials. However, our results are only for , where is an integer, .

We recall the definitions of the reverse Hölder class (e.g., [3]). We denote by the ball centered at with radius .

Definition 1 (reverse Hölder class). Let . For one says that if and there exists a positive constant such that
holds for every and . One says that if and there exists a positive constant such that
holds for every and .

Remark 2. If is a polynomial and , then belongs to (see [10, page 146]). For , it is easy to see .

Definition 3 (see [3, Definition 1.3]). (1) Let and . Then it is well known that there exists a positive such that (see [11, Lemma 2]). Then the function is well defined by
and satisfies for every .

Remark 4. If , , then there exists a positive constant such that
(cf. [3, Lemma 1.8] and [5, Lemma 2.2 (a)]). If then there exists a positive constant such that (see [3, Remark 2.9]).Let denote the multi-index with , , and . Define and for . For any positive integer and a function , denote that and . We denote by the fundamental solution for . The operator is the integral operator with as its kernel.

Now we state our theorems.

Theorem 5. Let , , and be integers, , , and . Suppose that for some . Then there exist positive constants such that
where .

Remark 6. In Theorem 5, for the case , inequality (6) was shown in [3, Remark 2.9] under the assumption . If we take the limit , then class becomes and implies “ and .” See [8, Remark 3] and also [8, Theorem 1].

Remark 7. We assume “ and .” Then inequality (6) is true for . See Section 7 for the details. We also remark that the inequality (6) is true for and under the assumption “ and .” See [3, Remark 2.9] and also [8, Theorem 1 (2)].

Theorem 8. Let , , and be integers, , , and . Suppose that for some , . Then there exist positive constants such that (6) holds, where and .

To prove Theorems 5 and 8 we need the estimates of the fundamental solution (Theorems 9 and 10). The following Theorem 9 generalizes the results in [2, Theorem 3.1] to the operator with potential which belongs to the reverse Hölder class.

Theorem 9. Let and be integers, , and . Suppose that . Then for any positive integer there exists a positive constant such that

In Theorem 9, the cases and were shown in [3, Theorem 2.7] and [8, Theorem 2], respectively. We prove Theorem 9 by induction; that is, we assume that Theorem 9 is true for and show the case . We also prove the following theorem which states derivative estimates of the fundamental solution.

Theorem 10. Let , , and be integers, , , and . Suppose that . Then for any positive integer there exists a positive constant such that

In Theorem 10, the cases and were shown in [3, page 537] and [8, Theorem 6], respectively.

The plan of this paper is as follows. In Section 2, we describe some lemmas needed later. In Section 3, we assume that Theorem 9 is true for and show some estimates for which are needed to prove the case in Theorem 9. In Section 4, we prove the case in Theorem 9. Section 5 is devoted to proof of Theorem 10. In Section 6, we prove Theorems 5 and 8. Finally, in Section 7, we state some remarks. Throughout this paper the letter stands for a constant not necessarily the same at each occurrence.

2. Preliminaries

In this section, we describe some lemmas needed later. First, we remark that , , implies that is a doubling measure; that is, there exists a positive constant such that
(see [3, page 518] and also [5, Remark 1.2 (1)]).

Lemma 11 (see [3, Lemma 1.2]). Assume that for some . Then there exists a positive constant such that, for ,

Lemma 12 (see [3, Lemma 1.4 (b), (c)]). Suppose that . Then there exist positive constants and such that, for ,
Suppose . Then there exist positive constants and such that, for , ,

Lemma 13 (Caccioppoli type inequality). Let , , and be integers, , , and . Assume that in for some . Then there exists a positive constant such that

For readers’ convenience, we give the proof of Lemma 13 at the end of this section. If is an even number, then letting in Lemma 13, we have the following.

Corollary 14 (see [2, Lemma 3.5]). Let be an integer, an even number, , and . Assume that in for some . Then there exists a positive constant such that

Lemma 15 (Caccioppoli type inequality). Let , , and be integers, , , and . Assume that in for some . Then there exists a positive constant such that

Corollary 16 (see [2, Lemma 3.5]). Let be an integer, an odd number, , and . Assume that in for some . Then there exists a positive constant such that

Lemma 17. Let , , and be integers, , , and . Assume that , , in for some . Then there exists a positive constant such that

In Lemma 17, the cases and have been proved (see [3, page 523] and [1, Corollary 5.6]).

Proof of Lemma 17. We proceed following the proof of [1, Corollary 5.6]. We prove Lemma 17 for which satisfies , , in . Let and be nonnegative integers and choose such that on and . Let be the fundamental solution for . Note that
where
and the summation is taken over all integers and satisfying , , and . Then, integrating by parts, for we have
Let
Then we have
It remains to estimate . Since , it follows that . Hence is subharmonic. Then for all ,
Then we have
Then using Lemmas 13 and 15 repeatedly, we have
It follows that
From (26) we have, for all ,
Then we arrive at the desired estimate.

Lemma 18. Let , , and be integers, , , and . Suppose that for some , . Assume also that in for some . Then there exists a positive constant such that
where .

In Lemma 18, the cases and were shown in [3, Lemma 4.6] and [8, Lemma 7], respectively.

Proof of Lemma 18. We show Lemma 18 by a method similar to the one used in the proof of [3, Lemma 4.6]. Let such that on and , where and are nonnegative integers. We denote by the fundamental solution for . Note that
where the summation is taken over all integers and satisfying , , and . Then integrating by parts, for , we have
It then follows from the well-known theorem on fractional integrals that
where and we have used Remark 4 (1).

At the end of this section, we give the following.

Proof of Lemma 13. Let and be nonnegative integers and choose such that on and . Multiplying by and integrating over by integrating by parts, we have
where the summation is taken over all integers and satisfying , , and . Let be a positive real number which will be determined later. Then the right-hand side of (32) is bounded by
Then choosing such that , we arrive at the desired estimate.

3. Estimates for

In this section, we assume that Theorem 9 is true for and show estimates for which is needed to prove the case in Theorem 9.

Lemma 19. Let , , and be integers, , and . Suppose that . Assume also that Theorem 9 is true for . Then there exists a positive constant such that
where .

Proof. We can prove Lemma 19 by the same way as in the proof of [3, Corollary 2.8]. It follows from the inductive assumption that
If , then
Let . Then using Hölder's inequality and (35) we have
It follows that
Finally, note that, by the inductive assumption and Lemma 12 (1),
if we choose . Then Lemma 19 follows.

We also need the following lemma.

Lemma 20. Let , , and be integers, , , and . Suppose that for some , . Assume also that Theorem 9 is true for . Then for there exists a positive constant such that
where .

Proof. We show Lemma 20 by a method similar to the one used in the proof of [3, Theorem 4.13]. Suppose that for some . Then for some , . Let
The adjoint of is given by
By duality, it suffices to show that
where . Let . We choose and such that , . Thus . By Hölder’s inequality we have
Let be a positive integer and . It follows from Lemma 18, Lemma 12 (1), and the inductive assumption that
where is a finite integer not depending on and . Thus
where we choose and is the Hardy-Littlewood maximal operator. Hence it follows that
Then (43) follows since .

Corollary 24. Let , , and be integers, , , and . Suppose that . Assume also that in for some . Then there exists a positive constant such that

Now we are ready to give the following.

Proof of Lemma 22. Let such that on , , where is an integer satisfying . Applying Lemma 23 to and using Corollary 14 we have
From Lemma 12 (2) it follows for each integer satisfying that
Then we have
Repeating the above argument, for any positive integer we have
Then using Lemma 17, Corollary 24, and estimate (56), we arrive at the desired estimate.

In this section, we prove Theorem 10 which states derivative estimates of the fundamental solution for . We arrive at Theorem 10 combining Lemmas 25 with 22.

Lemma 25. Let , , and be integers, , , and . Suppose that . Assume also that in for some . Then there exist positive constants and such that

Proof. Let such that on and , where and are nonnegative integers. We use (29); integrating by parts, we have
Since , it follows that for some . We choose such that and . By Hölder’s inequality we have
where we have used Remark 4 (1). From (59) we have, for all ,
Using Lemma 12 (1), we have
Then the proof is complete.

Proof of Theorem 5. We assume that for some . Then for some . Let
The adjoint of is given by
Let . By duality, it suffices to show that
where . Let . By Theorems 9 and 10 and Hölder's inequality we have
where . Since we have
For the case , by Lemma 11 and (4), we have
For the case , by the doubling condition (9) and (4), we have
where . Hence, choosing , we obtain
It follows that
Then (64) follows since .

Proof of Theorem 8. We assume that for some , . Then for some , . We use (62) and (63) again and it suffices to show (64) for and . We choose and such that and . Thus,
Let . By Hölder’s inequality we have